^' -i  O  l^'tOy 


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ate 


SO  cts, 
A 

GRAPBir  METT^l) 

FOR  SOLVING    CERTA1.>' 

ALGEBRAIC  PROlJLEMS 


GEORGE  "  .  VOPE; 


PROFESSOR   OF  CIVIL  ENGT'-vertnG  IN  BO"\V     ^11    COLLEGE, 
AUTHOR   OF     'MANUA:  MuROAD   EN(  INEERS/' 


UC-NRLF 


NEW    YOirr 
\    VAN    ISrOSTRAN^\     MTBLi       . 

23  Murray  Street  and  2",   V,       TvEN  S        kt. 
is  (  :). 


...I 


IN   MEMORIAM 
FLORIAN  CAJORI 


VAN  NOSTEAND'S  SCIENCE  SERIES. 


16. 
A   GRAPHIC   METHOD   FOR   SOLVING  CER- 
TAIN ALGEBRAIC  EQUATIONS.    By  Prot. 
George  L.  Yose.     With  Illustrations. 

IT. 
WATER    AND    WATER    SUPPLY.       By  Prof. 
W.  II.  CouFiELD,  M.A.,  of  the  University  Col- 
lege, London. 

18. 

SEWERAGE  AND  SEWAGE  UTILIZATION, 
By  Prof.  W.  II.  CoRFiELD,  M.A.,  of  the  Uni- 
versity College,  London. 

le. 

STRENGTH  OF  BEAMS  UNDER  TRANS- 
VERSE LOADS.  By  Prof.  W.  Allan,  author 
of  *'  Theory  of  Arches."     With  Illustrations. 

18  mo,  boards  50  cents  each. 


%*  Sent  free  by  mail  on  receipt  of  price. 


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FiGUEE  29,  p.  57. 


GEAPHIC  METHOD 


FOR  SOLVING  CERTAIN 


ALGEBRAIC  PROBLEMS 


GEOEGE  L.  yOSE, 

PROFESSOR  OP  CIVIL  ENGINEERING  iN  BOWDOIN  GOLLEOE, 
AUTHOR  OF  "manual  FOR  RAILROAD  ENGINEERS." 


NEW  YORK: 

D.  VAN  NOSTRAND,  PUBLISHER, 

23  Murray  and  27  Warren  Street. 

1  875. 


Copyright  1875,  by  D.  Van  Nostrand. 


^5-^ 


V4, 


PREFACE. 


A  portion  of  the  following  pages  first 
appeared  in  Van  Nostrand's  Engineer- 
ing Magazine  for  June,  1875.  The 
method  was  suggested  by  the  common 
mode  of  representing  the  movement  of 
railway  trains,  which  was  employed  as 
long  ago  as  1850,  and  was  first  brought 
to  the  writer's  knowledge  by  the  late 
S.  S.  Post,  the  well  known  Civil  Engin- 
eer. It  is,  of  course,  not  presented  as  in 
any  way  taking  the  place  of  the  far 
more  elegant  and  precise  methods  of 
analysis,  but  only  as  in  some  cases  a  con- 
venient mode  of  obtaining  a  bird's-eye- 
view  of  a  problem,  and  as  affording  the 
means  for   interpreting   certain   results. 


6 

which  by  other  processes  are  not  at  first 
sight  quite  plain. 

The  "  Cross  Section  Paper,"  employed 
by  engineers,  will  be  found  well  adapted 
for  the  working  of  problems  by  the 
graphic  method,  as  it  is  ruled  in  squares 
of  greater  or  less  size. 


A  GRAPHIC  METHOD 


rOR  SOLVING 


CERTAIN  ALGEB1|AIC  PROBLEMS 


The  various  methods  ordinarily  em- 
ployed for  the  solution  of  mathematical 
problems  are  well  known  to  all  who  are 
familiar  with  arithmetic,  algebra  and 
geometry.  There  is  however  a  method 
of  answering  a  certain  class  of  questions, 
and  of  representing  certain  results,  by  a 
direct  appeal  to  the  eye,  which  is 
extremely  simple,  very  effective  and  in 
some  cases  superior  to  every  other  mode. 
This  process  is,  at  least  in  some  of  its 
applications^  by  no  means  new  to  engin- 
eers, but  it  may  be  both  new  and  in- 
teresting to  some  persons,  and  it  is  pro- 
posed therefore  without  further  remarks 


to  present  a  few  examples  of  the  graphic 
method,  the  application  of  which  to 
additional  questions  will  readily  be 
made  by  the  reader. 

Suppose  we  have  the  following  ques- 
tion :  If  a  man  travels  five  miles  in  one 
hour,  how  far  will  he  go  in  four  hours. 
This  of  course  is  the  plainest  possible 
question  in  simple  multiplication.  But 
suppose  instead  of  the  above  we  have 
the  problem  below.  A  person  walked  a 
certain  distance  from  A  to  B  at  the  rate 
of  three  and  a  half  miles  an  hour,  and 
then  ran  a  part  of  the  way  back  from  B 
to  A,  at  the  rate  of  seven  miles  an  hour, 
walking  the  remaining  distance  in  five 
minutes,  and  being  out  twenty-five  min- 
utes in  all.  A  second  man  walks  from  B 
to  A  and  back  again,  at  a  uniform  rate^ 
being  also  out  twenty-five  minutes  in  all. 
At  what  two  times  will  he  meet  the  first 
man,  and  how  far  from  A  will  the  two 
points  of  meeting  be?  Here  now  is  a 
question  which  our  simple  multiplication 
will  not  answer  ;  but  by  the  graphic 
method  the  second  question  is  nearly  if 
not  quite  as  simple  as  the  first. 


9 

To  begin  with  our  first  question  above, 
draw  a  horizontal  line  and  divide  it  into 
equal    parts    as    at    1,  2,  3,  in  Fig.  1- 


c 

1 

2. 

3 

4 

5' 

/&' 

20 

V 

A 

\ 

\ 

B 

\ 

\ 

c 

^ 

a        /         z        ^        >w 
Fig.  1. 

Let  these  equal  horizontal  distances 
represent  hours.  Through  each  of  the 
points  0,  1,  2,  3  and  4  draw  the  vertical 
lines  0-0,  1-1,  2-2,  3-3  and  4-4,  and 
upon  the  first  vertical  line  lay  off  equal 
divisions  as  at  5,  10,  15  and  20,  to 
represent  miles,  and  through  the  points 
draw  lines  parallel  to  the  upper  horizontal. 
We  have  here  time  laid  off  upon  one 
line,  and  distance  laid  off  upon  another 
line  at  right  angles  to  the  first.  Now  if 
the  man  travels  five  miles  in  one  hour, 
his  path  is  represented  upon  our  diagram 


10 

by  the  diagonal  line  from  0  to  A;  any 
inclined  line  in  the  figure  representing  a 
movement  both  in  space  and  time.  If 
we  wish  to  know  how  far  the  man  will 
go  in  two  hours  we  have  only  to  draw  a 
vertical  through  2  to  cut  the  diagonal  at 
B,  and  from  B  to  draw  a  horizontal  line 
to  our  vertical  scale  of  miles  at  10;  or  if 
we  wish  to  know  how  long  the  man  will 
be  in  going  fifteen  miles  we  draw  a 
horizontal  from  15  to  cut  the  diagonal 
at  C,  and  through  C  draw  a  vei^tical  to 
cut  the  time  line  at  3.  If  a  second  man 
goes  twice  as  fast  as  the  first,  his  path 
will  be  shown  by  the  more  steeply  inclin- 
ed line  from  0,  on  the  upper  horizontal, 
to  2  upon  the  lower  one,  which  passes 
through  the  intersection  of  one  hour  and 
ten  miles.  Suppose  the  question  was  as 
follows  :  Two  men  start  from  the  same 
point  at  the  same  time  one  going  at  the 
rate  of  five  miles  and  the  other  at  ten 
miles  an  hour;  how  far  apart  will  they 
be  at  the  end  of  two  hours  ?  We  see  at 
once  that  the  vertical  distance  between 
our  two  inclined  lines,   measured  upon 


11 

the  perpendicular  through  2,  is  the  dif- 
ference between  ten  and  twenty  miles, 
or  ten  miles.  Let  us  reverse  the  ques- 
tion, thus  :  Two  men  start  from  the 
same  point  at  the  same  time,  and  travel, 
one  at  the  rate  of  five  and  the  other 
at  the  rate  of  ten  miles  an 
hour;  after  a  certain  time  they  are  ten 
miles  apart;  how  long  have  they  been 
traveling  ?  Here  we  have  only  to  take 
our  distance  representing  ten  miles  and 
find  where  it  will  just  go  in  vertically  be- 
tween the  inclined  lines,  and  then  pro- 
duce it  upwards  till  it  cuts  the  time  line, 
which  in  this  case  is  at  2 ;  thus  showing 
that  they  have  been  traveling  two 
hours.  Suppose  again  that  our  first 
man  starts  from  a  certain  point,  and 
that  at  the  end  of  four  hours  he 
has  gone  twenty  miles.  A  second  man 
starts  from  the  same  point  at  the  same 
time  and  reaches  the  end  of  the  twenty 
miles  two  hours  sooner  than  the  first 
man;  how  fast  did  he  travel?  In  this 
case  we  have  only  to  go  back  upon  the 
horizontal  line  from  4  to  2,  and  draw  a 


12 
line  from  2  upon  the  lower  horizontal  to 
0  upon  the  upper  one;  the  inclination  of 
this  line  will  give  us  the   rate   required^ 
or  ten  miles  an  hour. 

The  above  questions  are  extremely 
simple,  so  simple  indeed  as  to  be  done 
in  the  head  by  any  member  of  a  common 
school,  but  they  illustrate  the  method, 
which  we  will  apply  directly  to  more 
difficult  problems. 

We  have  seen  that  differently  inclined 
lines  represent  different  rates  of  move- 
ment. Let  us  take  another  question  : 
X  starts  from  a  certain  point  and  tra- 
vels in  a  certain  direction  for  a  cer- 
tain time,  his  path  being  represented 
by  the  diagonal  A  B  in  Fig.  2. 


A 

C  l/l           F 

\ 

v 

X 

G 

>4^V 

/ 

, 

/A 

/ 

X       s' 

V 

c 

/ 

\ 

\ 

Fig.  2. 


13 

Y  starts   an   hour  later   and  passing 
over  the  same  distance   arrives  an  hour 
earlier.     How  fast  did  Y  go,  and  when 
and  where  did  he  pass  X  ?   The  line  C  D 
in  the  diagram  represents  the  movement 
of  Y,  its  inclination  shows   his  rate,  and 
he  passes  X  at  a  distance  represented  on 
the  vertical  scale  by  F  S,  and  at  the  time 
shown  by  F  upon  the   upper  horizontal. 
A  third  man,  Z,  starts  from  the  opposite 
end  of  the  course  at  the  same  time  that 
X  leaves  the  first  end,  and  goes   at  the 
rate  of  the   second  man,  Y;  when  and 
where  will  he  cross  the  paths  of  the  two 
other  men  ?    It  will  be  seen  that  while 
two  men  may  move  in  opposite  directions 
time  always  goes  in  the  same   direction, 
and  though  a  man   may   stand  still,   or 
even  retrace  his  steps,  time  always  goes 
on.     As  a  matter  or  convenience  time  is 
always  represented  as  going  from  left 
to     right,    in    a    horizontal     direction. 
The  movement  of  the   third   man   Z  is 
therefore  shown  by  the  line  E  F  and  he 
will  pass  X  at  M  on   the   scale  of  miles, 
and  at  the  time  represented  by  N.    He 


14 

will  also  pass  Y  on  the  second  horizon- 
tal for  distance,  and  half  fway  between 
C  and  F  for  time.  Let  us  change  the 
question  with  regard  to  Z,  thus:  Z  leaves 
the  second  end  of  the  route  at  the  same 
time  that  X  leaves  the  first  end,*but  tra- 
vels twice  as  fast  until  he  has  gone  half 
the  length  of  the  course,  when  he  stops 
until  Y  overtakes  X  and  then  goes  on 
arriving  at  X's  starting  point  at  the 
same  time  that  X  arrives  ^at  his  (Y's) 
starting  point.  What  is  Z's  rate  during 
the  last  half  of  his  course  ?  In  this  case 
the  first  half  of  Z's  course  is  represented 
by  the  lineE  X;  but  as  he^now  stops  for 
an  hour  we  pass  along  on  the  horizontal 
from  X  to  S.  The  remainder  of  his 
course  is  shown  by  the  diagonal  from  S 
to  T,  the  inclination  of  which  is  evident- 
ly the  same  as  that  of  S  B.  The  rate  of 
Z  therefore  during  the  last  half  of  his 
course  is  the  same  as  the  uniform  rate  of 
X. 

The  various  algebras  and  arithmetics 
abound  in  questions  like  the  following  : 
Edinburgh   is   360   miles  from  Londo:.. 


15 


A  starts  from  Edinburgh  and  travels  at 
the  rate  of  10  miles  an  hour;  B  starts 
from  London  and  goes  eight  miles  an 
hour.  If  they  travel  towards  each  other 
when  and  where  will  they  meet  ? 
In   this  case   we    lay   off  EL,   Fig.    3, 


Fig.  3. 

equal  by  any  scale  to  360  miles.  Next 
laying  off  any  equal  parts  upon  the  line 
ED,  to  represent  hours,  we  draw  the 
diagonal  E  C  at  such  an  inclination  as  to 
show  the  rate  of  A,  viz.  ten  miles  an 
hour.  As  B  goes  in  the  opposite  direc- 
ion  the  diagonal  showing  his  movement 
will  be  inclined  as  by  the  line  L  D,  the 
angle  of  which  is  of  course  to  represent 


16 

the  speed  of  eight  miles  an,  hour. 
The  diagonals  cross  at  X,  from  which 
point  we  draw  X  M  and  X  N.  E  M  by 
our  vertical  scale  of  miles  will  be  the 
distance  from  Edinburgh,  and  N  upon 
the  time  line  will  show  the  time  at  which 
the  two  men  meet. 

Let  us  try  the  following  question.  A 
privateer  running  at  the  rate  of  ten 
miles  an  hour  sees  a  ship  eighteen  miles 
off  going  at  the  rate  of  eight  miles  an 
hour;  how  far  can  the  ship  go 
before  it  is  overtaken.     Let  AB,  Fig.  4, 


Fig.  4. 

represent  the  eighteen  miles  which  the 
ship  is  in  advance  of  the  privateer  when 
first  seen.     Also  let  AC   represent  the 


privateer's  rate,  or  ten  miles  an  hour, 
and  let  B  C  represent  the  rate  of  the 
ship,  or  eight  miles  an  hour. 
The  diagonals  produced  will  intersect  at 
C,  and  drawing  CD  and  C E  we  have 
A  D  for  the  time  and  B  E  as  the  distance 
which  the  ship  can  go  before  being  over- 
taken. 

Suppose  that  we  have  the  following 
question  :  Two  towns  are  fifty  miles 
apart,  A  is  to  leave  one  of  these  towns 
at  six  o'clock  and  to  arrive  at  the  other  at 
noon,  making  four  stops  of  half  an  hour 
each  at  ten,  twenty, thirty,and  forty  miles 
from  the  starting  point.  B  leaves  the 
other  end  of  the  road  at  seven  o'clock, 
travels  twenty  miles  an  hour  for  one 
hour,  then  turns  back  and  retraces  his 
course  for  an  hour  at  the  rate  of  ten 
miles  an  hour,  then  turns  around  and  ad- 
vances again  at  such  a  rate  as  to  meet  A 
as  he  is  starting  from  his  third  halt  ; 
continuing  at  the  same  rate  B  meets  at 
half  past  ten  a  third  man,  C,  who  left 
the  first  end  of  the  route  two  hours  later 
than  A  did  and  has  been  going  at  a  uni- 


18 

form  rate.  At  what  rate  has  C  been 
traveling,  and  where  did  B  meet  him  ? 
By  the  ordinary  process  this  question 
would  not  be  a  simple  one,  but  it  is 
quite  so  by  the  graphic  method,  as 
seen  by   the  diagram,  Fig.  5,  in  which 


Fig.  5. 

B  is  seen  to  meet  C  at  about  23  miles 
from  A's  starting  point,  and  C  is  found 
to  have  been  going  at  the  rate  of  about 
nine  and  a  half  miles  an  hour.  Our 
figure  is  too  small  to  give  the  required 
result  with  accuracy.  It  is  to  be  ob- 
served in  regard  to  all  of  these  pro- 
blems that  the  size   and   the   proportion 


19 


of  the  diagram  must  depend  entirely 
upon  the  degree  of  accuracy  which  it  is 
desired  to  obtain,  and  also  upon  the 
character  of  the  question.  Very  oblique 
cuttings  of  diagonals  should  be  avoided. 
Todhunter  gives  the  following  in  his 
elementary  algebra.  A  person  walked 
out  a  certain  distance  from  A  to  B  at 
the  rate  of  three  and  a  half  miles  an 
hour,  and  then  ran  part  of  the  way  back 
again  at  the  rate  of  seven  miles  an  hour, 
walking  the  remaining  distance  in  five 
minutes.  He  was  out  25  minutes  ;  how 
far  did  he  run  ?    Let  A  B,  Fig.  6,  repre- 


FiG.  6. 

sent  the  whole  time,  or  25  minutes. 
Lay  off  AH  equal  to  any  convenient 
fraction  of  an  hour,  and  A I  equal  to  the 


20 

corresponding  fraction  of  three  and  a 
half  miles :  the  diagonal  A  K  will  then 
by  its  inclination  represent  the  rate  of 
three  and  a  half  miles  an  hour;  produce 
this  diagonal  indefinitely  toward  C. 
Next  lay  off  B  L  equal  to  five  minutes 
upon  the  time  scale,  draw  the  vertical 
LM,  and  the  diagonal  BD  inclined  at 
the  same  rate  as  the  line  A  K.  Finally 
from  D  draw  the  diagonal  D  C  inclined 
at  such  a  rate  as  to  represent  seven  miles 
an  hour,  upon  the  same  scales  of  course 
as  A  C  represents  three  and  a  half  miles 
an  hour,  and  produce  it  to  intersect  A  C 
at  C.  The  whole  distance  between  the 
two  points  is  then  shown  by  B  F,  and  the 
distance  which  the  man  ran  by  D  M  or 
E  F,  measured  of  course  by  the  same 
scale  of  miles  before  employed. 

Suppose  to  the  preceding  question  we 
add  the  following  :  While  the  man  above 
referred  to  walks  from  A  to  B,  and  runs 
and  walks  back  again,  a  second  man 
walks  from  B  to  A  and  back  again  from 
A  to  B,  at  a  uniform  rate,  being  occu- 
pied in  all  the  same  length   of  time  as 


21 

the  man  first  mentioned;  at  what  points 
and  at  what  times  will  he  meet  the  first 
man  ?    We  will  repeat  in   Fig.  1    the 


lines  showing  the  movement  of  the  first 
man,  viz.  A  C,  C  D  and  D  B.  A  B  rep- 
resents the  whole  time  as  before,  and 
A  E  the  distance  between  the  two  points; 
then  will  E  F  and  F  G  represent  the 
movement  of  the  second  man,  and  he 
will  meet  the  first  man  on  his  outward 
trip  at  a  distance  from  his  starting  point 
shown  by  A  I,  and  after  the  time  A  H^ 
and  on  his  inward  trip  at  a  distance  B  K, 
and  at  the  time  A  J. 

The  question  below  is  also  given  in  the 
work  above  referred  to  :  A  person  walk- 


22 


ed  out  from  Cambridge  to  a  village  at 
the  rate  of  four  miles  an  hour,  and  on 
reaching  the  railway  station  had  to  wait 
ten  minutes  for  the  train,  which  was  then 
four  and  a  half  miles  off.  On  arriving 
at  his  rooms,  which  were  a  mile  from  the 
Cambridge  station,  he  found  that  he  had 
been  out  three  and  a  fourth  hours.  Find 
the  distance  of  the  village  from  Cam- 
bridge. In  this  case  we  first  lay  off  A  B, 
Fig.  8,  equal  by  any  scale  to  three  and  a 


D  E 


Fig.  8. 


fourth  hours.  We  next  make  A  L  equal 
to  one  hour,  and  A  M  equal  to  four  miles, 
when  the  diagonal  A  N  represents  the 


23 

\ 
rate  of  four  miles  an  hour,  which  we 
produce  indefinitely.  Next  we  go  back 
from  B  to  C  the  five  minutes  which  it 
takes  the  man  to  go  from  the  Cambridge 
station  to  his  rooms,  and  draw  the  line 
C  E,  representing  the  rate  of  the  railway- 
train,  and  produce  it  indefinitely.  If 
the  man  had  not  been  obliged  to  wait  for 
the  train  we  should  simply  produce  the 
two  diagonals  until  they  met,  when  the 
vertical  distance  of  their  intersection  from 
the  upper  horizontal,measuredonthe  scale 
of  miles,  would  be  the  distance  required. 
As,  however,  the  man  has  to  wait  ten 
minutes  at  the  station,  we  take  the  dis- 
tance D  E  equal  to  that  time,  and  find 
where  it  will  just  go  in  horizontally  be- 
tween the  two  diagonals,  when  the  ver- 
tical distance  between  D  E  and  A  B  will 
be  what  we  require.  If  the  whole  time 
being  thesame  the  man  had  waited  an  hour 
at  the  station,  and  we  wished  to  know  the 
distance,  we  should  apply  the  line  H  I, 
equal  to  one  hour  by  the  time  scale,  to 
the  diagonals,  and  K  P  would  give  us  the 


24 

distance  ;  or  if  the   distance   K  P  was 
given  we  should  obtain  the  time  H  L 

Let  us  now  pass  to  a  somewhat  different 
class  of  questions  :  Two  men  start  at 
the  same  time  to  walk  round  an  island  ; 
the  first  man  goes  at  the  rate  of  five 
miles  an  hour  ;  the  speed  of  the  second 
man  is  such  as  to  carry  him  round  the 
island  in  three  and  a  third  hours,  the  dis- 
tance being  ten  miles.  How  long  after 
starting  will  the  first  man  pass  the  sec- 
ond, and  how  long  before  he  will  pass 
him  the  second  time  ?  The  reader  will, 
perhaps,  at  first  sight  not  see  the  relation 
between  movement  on  the  circular  path 
and  time,  as  it  is  a  little  different  from 
the  relation  between  movement  on  a 
straight  line  and  time.  He  has,  however, 
only  to  observe  that  in  traveling  a  circu- 
lar path  a  man  while  always  getting 
farther  away  from  the  starting  point  is 
at  the  same  time  getting  nearer  to  it, 
or,  in  other  words,  he  is  traveling  both 
from  it  and  towards  it  at  the  same  time. 
Our  question  above  thus  takes  the  form 
shown  in  Fig.  9,  in  which  the  movement 


25 


of  the  first  man  is  shown  by  the  diagon- 
als AB,  CD,  E  F,  etc.,  and  that  of  the 
second  man  by  the  dotted  diagonals.  It 
will  be  seen  that  having  drawn  A  B  we 
recommence  at  C  ;  this  is  because  in 
going  from  the  point  represented  by  the 
upper  horizontal  line  to  the  point  repre- 
sented by  the  lower  horizontal,  inasmuch 
as  the  path  is  a  circular  one,  we  have  got 
back  again  to  the  starting  point.  The 
first  man  it  will  be  seen  passes  the  sec- 
ond at  five  hours  after  starting,  and 
again  at  ten  hours.  K,  instead  of  both 
going  from  A  towards  N",  one  of  the 
men  goes  from  N  towards  A,  we  have 
only  to  start  from  the  lower  line  and  in- 


26 

cline  the  diagonal  in  the  opposite  direc- 
tion, and  we  may  vary  the  rates  of  speed, 
and  stop  the  men  at  any  points,  for  any 
length  of  time,  without  making  the 
question  any  more  difficult.  For  ex- 
ample, the  movement  of  a  man  who 
should  travel  in  the  opposite  direction  at 
the  rate  of  one  mile  an  hour  is  shown  by 
the  diagonal  N  O,  and  he  will  meet  the 
second  of  the  men  above  referred  to  at 
P,  Q  and  E,  from  which  points  we  may 
draw  verticals  to  the  time  line,  and  hor- 
izontals to  the  line  A  N,  which  will  show 
us  just  when  and  where  the  several  meet- 
ings will  take  place. 

We  find  the  following  question  in  Tod- 
hunter's  Algebra  :  A  and  B  start  to- 
gether from  the  same  point  on  a  walking 
match  round  a  circular  course.  After 
half  an  hour  A  has  walked  three  com- 
plete circuits,  and  B  has  walked  four  and 
a  half  ;  assuming  that  each  walks  with 
uniform  speed  find  when  B  overtakes  A. 
Let  A  B,  Fig.  10,  represent  the  length  of 
the  course,  and  let  A  C  or  B  H  represent 
half    an  hour :    then  the    dotted    line 


A  D  E  F  G  H  will  show  the  movement  of 
A,  while  the  four  and  a  half  full  diagon- 
als to  I  will  show  that  of  B.  Carrying 
the  two  sets  of  lines  on  at  the  same 
rate  we  find  them  together  again  at  J, 
which  by  the  time  scale  is  ten  minutes 
from  the  time  represented  by  H. 

Let  us  try  some  of  the  watch  problems 
as  given  in  the  algebras.  In  Fig.  11  we 
have  shown  the  movement  of  both  the 
hour  and  minute  hands  for  twelve  hours, 
and  we  shall  find  that  the  several  diag- 
onals   answer    a  variety  of    questions. 


28 


Ml      2     3     4.    5     e      7    8     a     10     II    M 


Fig.  11. 

We  may  take  the  distance  around  the 
face  of  the  watch  as  representing  time 
or  distance  as  we  please.  It  represents 
both,  and  thus  we  lay  off  twelve  divi- 
sions upon  the  upper  horizontal  line  and 
also  upon  the  left  hand  vertical.  The 
long  diagonal  represents  the  course  of 
the  hour  hand  for  twelve  hours,  and  the 
short  diagonals  represent  the  twelve 
revolutions  of  the  minute  hand  in  the 
same  time.  Take  now  the  following 
question  :  The  hands  of  the  watch  are 
together  at  noon,  when  are  they  next  to- 
gether ?  We  see  plainly  that  the  hands 
are  together  at  noon,  at  a  little  after  one 
o'clock,  at  a  little  more  after  two  o'clock, 


29 

at  a  still  longer  time  after  three,  and  so 
on,  the  precise  time  being  found  by  car- 
rying the  crossings  of  the  diagonals  ver- 
tically upwards  to  the  time  line.  Our 
figure  is  too  small  to  do  this  accurately. 
Let  us  take  one  hour  out  of  the  preced- 
ing diagram,  and  enlarge  it,  as  in  Fig.  12. 


iVI       l\l 


W 
^ 


Fig.  12. 


Take  the  following  question :  The 
hands  of  a  watch  are  at  right  angles  at 
three  o'clock  ;  When  are  they  next  at 
right  angles?  The  hands  are  at  right 
angles  when  they  are  fifteen  minutes 
apart.     The  vertical  divisions  in  Fig.  12 


30 

are  each  five  minutes.     The  movement 
of  the  minute  hand  for  an  hour  is  shown 
by  the  diagonal  AB,   and  that  of  the 
the  hour  hand  by  C  D.     Wherever  we 
can  get  a  vertical  equal  to  fifteen  min- 
utes, or  to  A  C,  between  the  two  diagon- 
als the  hands  will  be  at  right  angles,  and 
by  producing  this  vertical   to  the  time 
line,  as  at  M,  we  get  the  required  time, 
in  the  present  case  between  thirty-two 
and    thirty-three     minutes     past    three 
o'clock.     Again,  let  the  question  be  to 
find  at   what   time  between   three   and 
four  o'clock  the  hands  will  be  diametri- 
cally  opposite.     Diametrically  opposite 
is  thirty  minutes   apart,   and   applying 
thirty  minutes,   or   six   of   the  vertical 
spaces,  to  the  lines  AB  and  CD,  and 
producing  the  line  upwards,  we  find  the 
time  line  to  be  cut  at  N,  or  about  eleven 
minutes  before  four  o'clock. 

It  will  be  evident  from  an  examina- 
of  the  preceding  figures  that  the  graphic 
method  is  not  confined  to  questions  in- 
volving time  and  space  alone,  but  that  it 
is  equally  applicable  to  questions  of  time 


/ 


31 

and  any  kind  of  work  done,  whether 
labor  performed  by  men,  water  discharg- 
ed by  pipes,  or  the  like.  Take  for  ex- 
ample the  following  question  :  A  can  do 
a  piece  of  work  in  five  days,  and  B  can 
do  it  in  three  days.  In  what  time  will 
both  working  together  do  it  ?  Let  A  B, 
Fig.  13,  be  the  time  in  which  A  can  do 


Fig.  13. 

an  amount  of  work  represented  by  A  C 
or  B  D,  then  will  A  D  represent  the  rate 
at  which  he  works.  So,  too,  if  B  does 
an  amount  of  work  shown  by  B  E,  in  the 
time  AB,  AE  will  represent  his  rate  of 


32 

work.  Make  EF  equal  to  B  D.  BF 
will  show  the  amount  of  work  done  by 
both  in  the  time  A  B,  and  A  F  will  be 
the  rate  at  which  both  together  work. 
The  several  rates  being  fixed  the  ques- 
tion is  at  once  answered.  The  amount 
of  work  being  represented  by  AM,  A 
will  do  it  in  the  time  represented  by  A  H, 
B  in  a  time  shown  by  A  J,  and  both  to  • 
gether  in  a  time  A  L.  The  same  diagram 
will  of  course  answer  other  questions  in 
regard  to  the  two  men.  For  example,  if 
we  know  the  time  in  which  both  men 
can  do  a  piece  of  work,  and  also  the 
time  in  which  one  man  can  do  it,  we  find 
easily  how  long  the  other  will  be  doing 
it. 

Questions  like  the  following  are  com- 
mon in  arithmetic  and  algebra:  A  syphon 
would  empty  a  cistern  in  forty-eight 
minutes,  while  a  cock  would  fill  it  in 
thirty-six  minutes.  When  it  is  empty 
both  begin  to  act.  How  soon  will  the 
cistern  be  filled  ?  Of  course  the  capa- 
city of  the  cistern  in  this  question  is  im- 
material ;  assume  it  to  be  AE,  Fig.  14. 


Fig.  14. 

The  syphon  can  empty  it  in  forty-eight 
minutes.  Lay  off  therefore  by  any  scale 
A  C  equal  to  forty-eight.  The  cock  can 
fill  it  in  thirty-six  minutes.  Lay  off 
A  B  equal  to  thirty-six.  The  diagonal 
A  H  will  represent  the  rate  at  which  the 
syphon  empties  the  cistern,  wliile  the 
diagonal  A  F  shows  the  rate  at  which 
the  cock  fills  it.  The  difference  between 
the  two  is,  of  course,  the  rate  at  which 
the  cistern  is  filled.  Producing  therefore 
the  diagonals  until  they  are  separated  by 
a  vertical  distance  equal  to  the  assumed 
capacity  of  the  cistern,  A  E,  that  is,  I K, 
and  carrying  I  K  up  to  the  time  line  at 
D,  we  have  AD  as  the  time  in  which 
the  cistern  will  be  filled  by  the  joint  ac- 
tion of  the  syphon  and  the  cock. 


34 

Suppose  that  the  question,  instead  of 
'being  as  above,  had  been  as  follows  : 
A  syphon  and  a  cock  acting  together 
will  fill  a  cistern  in  144  minutes,  while 
the  cock  acting  alone  would  fill  it  in 
thirty-six  minutes?  how  long  would  it 
take  the  syphon  to  empty  it  ?  Lay  off 
A  D,  Fig.  14,  equal  to  144  minutes,  and 
A  B  equal  to  thirty-six  minutes.  Make 
A  E  equal  to  the  capacity  of  the  cistern, 
upon  any  scale.  Draw  E  H  parallel  to 
A  D,  and  B  F  perpendicular  to  the  same. 
Through  A  and  F  draw  a  diagonal,  and 
produce  it  to  intersect  a  vertical  through 
D  at  K.  Make  K  I  equal  to  A  E,  and 
clraw  I A  to  intersect  the  horizontal  E  H 
at  H.  From  H  erect  a  perpendicular  to 
cut  the  time  line  at  C.  AC  will  then  be 
the  time  in  which  the  syphon  alone  would 
empty  the  cistern. 

Let  us  change  the  question  again  as 
follows  :  A  syphon  would  empty  a  cis- 
tern in  forty  minutes,  while  a  cock  would 
fill  it  in  twenty-two  minutes.  Both  com- 
mence to  act,  but  after  fifty-three  min- 
utes the  cock  is  stopped  for  twenty-two 


35 

minutes,  and  then  flows  again  at  a  rate 
which  would  fill  the  cistern  in  one  hour 
and  fifty-six  minutes.  When  the  cock 
recommences  the  syphon  stops  working 
for  sixteen  minutes,  but  after  that  time 
the  cistern  commences  to  leak  at  a  rate 
which  would  empty  it  in  one  hour  and 
twenty-five  minutes.  How  long,  under 
the  new  conditions,  will  it  be  from  the 
beginning  before  the  cistern  will  be  half 
full  ?  As  the  cock  stops  after  fifty-three 
minutes,  which  time  is  represented  upon 
the  upper  horizontal  in  Fig.  15  by  the 


Fig.  15. 

distance  S  G  we  draw   A  B  horizontally 
and  from  B  draw  B  C  at   such   an  angle 


36 

as  to  represent  the  new  rate  at  which 
the  cock  supplies  water.  In  the  same 
manner  we  draw  D  E  to  show  the  stop- 
page of  the  syphon,  and  afterwards  E  F 
to  represent  the  rate  of  leakage.  The 
diagonals  B  C  and  E  F  must  then  be 
produced  until  the  included  vertical  F  C 
is  by  the  scale  equal  to  one  half  of  the 
capacity  of  the  cistern,  or  one  half  of 
S  T.  Finally  we  produce  C  F  to  cut  the 
time  line  at  K,  and  S  K  is  the  answer  to 
the  question. 

The  various  questions  in  Alligation 
may  be  worked  by  the  graphic  method. 
Suppose  for  example  that  a  man  would 
mix  one  kind  of  grain  worth  thirty  cents 
a  bushel  with  another  quality  worth 
eighty  cents,  so  as  to  make  sixty  bushels 
worth  50  cents  a  bushel  ;  how  much  of 
each  kind  must  he  take  ?  Lay  off  B  F 
in  Fig.  16  equal  by  any  scale  to  thirty 
cents,  B  H  equal  to  fifty  cents,  and  B  I 
equal  to  eighty  cents.  The  several  lines 
A  F,  A  H  and  A  I  will  represent  the 
rates  or  values  of  the  different  kinds  of 
grain.     Make  A  J  on  any  scale  equal  to 


37 


Fig.  16. 

the  required  number  of  bushels  in  the 
mixture.  Draw  a  vertical  J  K  to  meet 
A  H,  the  value  or  rate  of  the  mixture 
produced,  at  K.  Produce  A  F  indefinite- 
ly, and  from  K  draw  K  L  parallel  to  A I 
to  intersect  AF  produced  inL,  and  from 
L  draw  the  vertical  L M.  AM  will  show 
the  number  of  bushels  at  the  price  B  F, 
and  M  J  will  give  the  number  at  the 
price  B  I.  The  result  will  of  course  be 
the  same  if  we  produce  A  I  and  draw 
K  N  parallel  to  A  L  to  meet  it  at  N,  and 
drop  the  perpendicular  NO  on  to  the 
line  P  K. 

Let  us  take   a   question,  such  as   we 


38 


frequently  find,  like  the  following  :  A 
workman  was  hired  for  forty  days  at 
three  shillings  and  four  pence  for  every 
day  that  he  worked,  but  with  the 
condition  that  for  every  day  he  did  not 
work  he  was  to  forfeit  one  shilling  and 
four  pence  ;  he  received  £3  3s  4d;  how 
many  days  did  he  work.  Reduce  the 
several  amounts  to  the  common  unit  of 
pence  for  convenience  in  plotting  the 
work  upon  paper.     Make  A  B,  Fig.  17, 


Fig.  17. 

by  any  scale  equal  to  forty  days,  and 
make  BC  equal  to  what  he  would  have 
received  had  he  worked  all  of  the  time. 
JVlake  A  D  equal  to  what  he  would  have 
lost  had  he  worked  none  at  all,  and  draw 
C  A  and  D  B.  Find  where  the  line  E  F 
which  is  equal  by  the  scale   of  pence  to 


39 


the  whole  amount  he  received,  will  just 
go  in  vertically  between  A  C  and  D  B 
and  produce  it  upwards  to  K.  A  K  upon 
the  scale  of  time  will  then  show  the  num- 
ber of  days  he  worked,  and  K  B  the 
number  of  days  he  was  idle,  the  former 
being  twenty-five  and  the  latter  fifteen. 
If  we  wish  to  know  how  many  days  he 
should  work  in  order  that  his  earnings 
may  just  balance  his  loss,  we  have  only  to 
draw  from  the  point  H,  where  the  two 
diagonals  cross,  the  vertical  HL  when 
we  shall  have  A  L  as  the  number  of  days 
he  worked  and  L  B  as  the  number  he 
was  idle. 

Todhunter  gives  the  following  question 
in  one  of  his  works.  A  and  B  shoot  by 
turns  at  a  target.  A  puts  seven  bullets 
out  of  twelve  into  the  bulls-eye,  while  B 
puts  in  nine  out  of  twelve  ;  between 
them  they  put  in  thirty-two  bullets. 
How  many  shots  did  each  man  fire  ? 
Lay  off  A  B,  Fig.  18,  equal  upon  any  scale 
to  thirty- two,  the  whole  number  of  suc- 
cessful shots.  Next,  lay  off  seven  of  the 
thirty-two  divisions  from  A   to   H,  and 


40 


twelve  of  the  same  divisions  on  the  ver- 
tical line  from  A  to  I,  then  will  the  diag- 
onal A  M  represent  A's  rate  of  success. 
In  the  same  way  we  lay  off  nine  divisions 
from  B  to  K  and  twelve  divisions  from 
Bj^to  L,  and  B  N  will  represent  B's  rate 
of  success.  Produce  A  M  and  B  N  until 
they  meet  at  C,  and  from  C  draw  C  D 
perpendicular  to  AB.  CD  will  show 
the  number  of  shots  that  each  man  fired. 
If  A  made  no  successful  shots,  proceed, 
ing  in  the  same  way  we  should  lay  off  no 
divisions  upon  the  line  A  B,  and  twelve 
upon  A  E,  and  the  line  representing  the 


41 


rate  of  success  would  be  vertical,  or  in 
other  words  he  has  no  success.  In 
such  case  the  number  of  times  that  B 
would  have  to  fire  to  put  the  thirty-two 
balls  into  the  target  would  be  found  by- 
producing  B  C  to  cut  A  E  produced,  the 
length  of  A  E  thus  produced  showing  the 
number. 

About  the  time  that  the  Pacific  Rail- 
road was  opened  the  newspapers  passed 
around  the  following  question  :  Sup- 
pose that  it  takes  a  train  just  one  week 
to  run  the  whole  length  of  the  road,  and 
that  one  train  leaves  each  end  of  the 
road  each  morning,  how  many  trains  will 
a  person  meet  in  going  the  length  of  the 
road,  not  counting  the  train  which  arrives 
just  as  he  starts,  nor  the  train  that  starts 
just  as  he  arrives  ?  Let  the  six  vertical 
divisions  in   Fig.    19   represent   the  six 


.  J^iG,  19. 


42 

days  .  The  diagonal  from  the  bottom 
of  the  left  hand  vertical  to  the  top  of 
the  right  hand  one  may  show  the  move- 
ment of  the  train  running  through  in  one 
direction,  when  the  opposite  diagonals, 
eleven  in  number,  will  show  the  number 
of  trains  which  he  will  meet. 

To  what  has  preceded  we  add  a  few 
examples  for  practice  from  Todhunter's 
algebras.  In  some  cases  we  have  given 
the  diagrams  showing  the  form  which 
the  solution  will  take,  while  in  other 
cases  the  construction  of  the  figure  is  left 
to  the  reader.  The  ease  with  which 
many  questions  commonly  found  in  the 
books  will  be  answered  by  the  graphic 
method  will  depend  of  course  upon 
the  more  or  less  perfect  knowledge  of 
algebra  which  may  be  possessed. 

Two  plugs  are  opened  in  a  cistern 
containing  192  gallons  of  water  ;  after 
three  hours  one  of  the  plugs  becomes 
stopped,  and  the  cistern  is  emptied  by 
the  other  in  eleven  more  hours  :  had 
six  hours  elapsed  before  the  stoppage  it 
would  have  required  only  six  hours  more 


43 


to  have  emptied  the  cistern.  How  many- 
gallons  will  each  hole  discharge  in  an 
hour,  supposing  the  discharge  uniform. 
In   Fig,  20   A  B  represents  three  hours, 


Fig.  20. 

B  C  three  hours,  C  D  six  hours,  and  D  E 
two  hours.  A  K  shows  the  discharge  by 
the  two  plugs  for  three  hours,  and  K I 
the  discharge  by  one  plug  for  the  eleven 
hours  additional.  The  inclination  of  the 
lines  must  be  such  that  when  A,  K  and 
L  are  in  a  straight  line,  L  H  and  K  I 
shall  be  parallel.  The  inclination  A  L 
will  show  the  rate  of  discharge  of  both 


44 

plugs,  and  K I  or  A  M  will  show  the  rate 
of  one,  while  the  difference  between  these 
two  or  A  N  will  show  the  rate  of  the 
other. 

The  road  from  a  place  A  to  a  place  B 
first  ascends  for  five  miles,  is  then  level 
for  four  miles,  and  afterwards  descends 
for  six  miles  the  rest  of  the  distance.  A 
man  walks  from  A  to  B  in  three  hours 
and  fifty-two  minutes;  the  next  day  he 
walks  back  to  A  in  four  hours,  and  he 
then  walks  half  way  back  to  B  and  back 
again  to  A  in  three  hours  and  fifty-five 
minutes.  Find  his  rates  of  walking  up 
hill,  on  level  ground  and  down  hill. 

A  and  B  are  two  towns  situated  24 
miles  apart  upon  the  same  bank  of  a  river. 
A  man  goes  from  A  to  B  in  seven  hours 
by  rowing  the  first  half  of  the  distance, 
and  walking  the  second  half.  In  return- 
ing he  walks  the  first  half  at  three-fourths 
of  his  former  rate,  but  the  stream  being 
with  him  he  rows  at  double  his  rate  in 
going,  and  he  accomplishes  the  whole 
of  the  distance  in  six  hours.  Find  his 
rates  of  walking  and  rowing. 


45 

A  and  B  set  out  to  walk  together  in 
in  the  same  direction  round  a  field,  which 
is  a  mile  in  circumference,  A  walking 
faster  than  B.  Twelve  minutes  after  A 
has  passed  B  for  the  third  time,  A  turns 
and  walks  in  the  opposite  direction  until 
six  minutes  after  he  has  met  him  for  the 
third  time,  when  he  returns  to  his  orig- 
inal direction  and  overtakes  B  four  times 
more.  The  whole  time  since  they  start- 
ed is  three  hours,  and  A  has  walked 
eight  miles  more  than  B.  A  and  B  di- 
minish their  rates  of  walking  by  one  mile 
an  hour,  at  the  end  of  one  and  two  hours 
respectively.  Determine  the  velocities 
with  which  they  began  to  walk. 

A  vessel  can  be  filled  with  water  by 
two  pipes  ;  by  one  of  the  pipes  alone  the 
vessel  can  be  filled  two  hours  sooner  than 
by  the  other  ;  also  the  vessel  can  be 
filled  by  both  pipes  together  in  labours. 
Find  the  time  which  each  pipe  alone 
would  take  to  fill  the  vessel.  The  dia- 
gram given  upon  a  preceding  page.  Fig. 
13,  is  the  same  as  that  required  for  the 
above  question,  by  which  the  answer  will 
be  seen  to  be  three  and  five  hours. 


46 

A  offers  to  run  three  times  round  a 
course  while  B  runs  twice  round,  but  A 
only  gets  150  yards  of  his  third  round 
finished  when  B  wins.  A  then  offers  to 
run  four  times  round  for  B's  thrice,  and 
now  runs  four  yards  in  the  time  he  for- 
merly ran  three  yards.  B  also  quickens 
his  rate  so  that  he  runs  9  yards  in  the 
time  he  formerly  ran  8  yards,  but  in  the 
second  round  falls  off  to  his  original  pace 
in  the  first  race,  and  in  the  third  round 
goes  only  9  yards  for  10  he  went  in  the 
first  race,  and  accordingly  this  time  A 
wins  by  180  yards.  Determine  the 
length  of  the  course. 

A  boat's  crew  row  3^  miles  down  a 
river  and  back  again  in  an  hour  and  40 
minutes.  Supposing  the  river  to  have  a 
current  of  2  miles  an  hour,  find  the  rate 
at  which  the  crew  would  row  in  still 
water. 

A  and  B  start  together  from  the  foot 
of  a  mountain  to  go  to  the  summit.  A 
would  reach  the  summit  half  an  hour  be- 
fore B,  but,  missing  his  way,  goes  a  mile 
and  back  again  needlessly,  during  which 


47 

he  walks  at  twice  his  former  pace,  and 
reaches  the  top  six  minutes  before  B.  C 
starts  twenty  minutes  after  A  and  B, 
and  walking  at  the  rate  of  two  and  one- 
seventh  miles  per  hour,  arrives  at  the 
summit  ten  minutes  after  B.  Find  the 
rates  of  walking  of  A  and  B,  and  the 
distance  from  the  foot  to  the  summit  of 
the  mountain. 

A  and  B  are  set  to  a  piece  of  work 
which  they  can  finish  in  thirty  days 
working  together,  and  for  which  they 
are  to  receive  <£7,10s.  When  the  work 
is  half  finished,  A  stops  working  for 
eight  days  and  B  for  four  days,  in  con- 
sequence of  which  the  work  occupies  five 
and  a  half  days  more  than  it  would 
otherwise  have  done.  How  much  ought 
A  and  B  respectively  to  receive  ? 

A  body  of  troops  retreating  before 
the  enemy,  from  which  it  is  at  a  certain 
time  26  miles  distant,  marches  18  miles 
a  day.  The  enemy  pursues  it  at  the  rate 
of  23  miles  a  day,  but  is  first  a  day  later 
in  starting,  then,  after  two  days'  march, 
is  forced  to  halt  for  one  day  to  repair  a 


48 


bridge,  and  this  they  have  to  do  again 
after  two  days  more  marching.  After 
how  many  days  from  the  beginning  of 
the  retreat  will  the  retreating  force  be 
overtaken?    AB,  in  Fig.  21,  represents 


A   C      1?  15      J  31 


Fig.  21. 

26  miles,  A  C  one  day,  C  D  two  days, 
D  E  one  day,  E  F  two  days,  and  F  H  one 
day.  The  inclination  of  the  line  B  L 
represents  the  rate  at  which  the  retreat- 
ing troops  march,  and  the  inclination 
of  C  M  or  M  N  or  N  L  shows  the  rate  of 
the  pursuers.  The  horizontals  M  and  N 
are  the  two  halts  of  a  day  each.  The 
two  diagonals  are  to  be  produced  until 
they  cut,  as  at  L,  when  A  K  will  give  us 


49 

the  time,  and  AG  or  K L  the  distance 
required. 

A  rows  at  the  rate  of  8j  miles  an  hour. 
He  leaves  Cambridge  at  the  same  time 
that  B  leaves  Ely,  and  is  back  in  Cam- 
bridge 2  hours  and  20  minutes  after  B 
gets  there.  B  rows  at  the  rate  of  7^ 
miles  an  hour,  and  there  is  no  stream. 
Find  the  distance  from  Cambridge  to 
Ely.     The  distance  A  B,  Fig.  22,  must 


be  such  that  when  AD  represents  8j 
miles  an  hour,  DE  12  minutes,  E  F  8  J 
miles  an  hour,  and  B  C  7  J  miles  an  hour, 
CF  shall  be  equal  to  2  hours  and  20 
minutes. 

Two  workmen  A  and  B  are  employed 


50 

by  the  day  at  different  rates.  A  at  the 
end  of  a  certain  number  of  days  receiv- 
ed £4  16s.,  but  B,  who  was  absent  six  of 
the  days,  received  only  £2  14s.  If  B 
had  worked  the  whole  time,  and  A  had 
been  absent  the  six  days,  they  would 
both  have  received  the  same.  Find  the 
number  of  days,  and  what  each  was  paid 
per  day.     AC,   in   Fig.   23,  shows   the 


Fig.  23. 

whole  time,  B  C  being  six  days.  C  E  is 
equal  to  £4  16s.,  and  B  K  to  £2  14s. 
A  B  must  be  such  that  T>  I  being  drawn 
parallel  to  A  C  the  lines  drawn  through 
E  and  I  and  through  D  and  K  shall  meet 
upon  the  line  A  B. 
A  waterman  rows  thirty  miles  and 


51 

back  in  twelve  hours,  and  he  finds  that 
he  can  row  five  miles  with  the  stream  in 
the  same  time  as  three  against  it.  Find 
the  times  of  rowing  up  and  down. 

A  person  hired  a  laborer  to  do  a  cer- 
tain work  on  the  agreement  that  for 
every  day  he  worked  he  should  receive 
2s.,  but  that  for  every  day  he  was  absent 
he  should  lose  9d.  ;  he  worked  twice  as 
many  days  as  he  was  absent,  and  on  the 
whole  received  £1  19s.  How  many  days 
did  he  work?    In  Fig.  24,  AC  is   to  be 


Fig.  24. 

twice  as  great  as  C  B,  A  D  is  to  represent 
the  man's  rate  of  receipt  while  he  work- 
ed, and  B  E  his  rate  of  loss  while  idle. 


52 


E  D  is  to  be  equal  by  the  scale  to  £1 
19s. 

A  man  and  a  boy  being  paid  for  cer- 
tain days'  work,  the  man  received  27 
shillings,  and  the  boy,  who  had  been  ab- 
sent three  days  out  of  the  time,  received 
twelve  shillings.  Had  the  man  instead 
of  the  boy  been  absent  three  days  they 
would  have  received  the  same  amount. 
Find  the  wages  of  each  per  day.  In 
Fig.  25,  A  E  represents  the  whole  time. 


Fig.  25. 

A  B  the  man's  rate  of  work,  E  B  what 
he  received,  AF  three  days,  FD  the 
boy's  rate  of  work,  and  E  D  what  the 
boy  received;  A  C  parallel  to  F  D,  shows 


,      63 

the  boy's  work  for  the  whole  time,  and 
F  C,  parallel  to  A  B,  the  man's  work 
omitting  three  days.  The  inclinations  of 
the  lines  must  be  such  that  F  C  and  A  C 
parallel,  respectively,  to  A  B  and  F  D 
shall  meet  on  E  B. 

A  railway  train  after  traveling  for  one 
hour  meets  with  an  accident  which  de- 
lays it  one  hour,  after  which  it  proceeds 
at  three-fifths  of  its  former  rate,  and  ar- 
rives at  the  terminus  three  hours  behind 
time  ;  had  the  accident  occurred  fifty 
miles  further  on,  the  train  would  have 
arrived  one  hour  and  twenty  minutes 
sooner.  Required  the  length  of  the  line, 
and  the  original  rate  of  the  train. 

The  fore  wheel  of  a  carriage  makes 
six  revolutions  more  than  the  hind  wheel 
in  going  120  yards;  if  the  circumference 
of  the  fore  wheel  be  increased  by  one- 
fourth  of  its  present  size,  and  the  cir- 
cumference of  the  hind  wheel  by  one- 
fifth  of  its  present  size,  the  six  will  be 
changed  to  four.  Required  the  circum- 
ference of  each  wheel. 

A  and  B  can  do  a  piece  of  work  to- 


54 

gether  in  48  days  ;  A  and  C  can  do  it  in 
30  days  ;  and  B  and  C  working  together 
can  do  it  in  26§  days.  Find  the  time  in 
which  each  could  do  the  work  alone. 

A  man  starts  from  the  foot  of  a  moun- 
tain to  walk  to  its  summit.  His  rate  of 
walking  during  the  second  half  of  the 
distance  is  half  a  mile  per  hour  less  than 
his  rate  during  the  first  half,  and  he 
reaches  the  summit  in  5^  hours.  He 
descends  in  3j  hours,  walking  at  a  uni- 
form rate  which  is  one  mile  an  hour 
more  than  his  rate  during  the  first  half 
of  the  ascent.  Find  the  distance  to  the 
summit,  and  his  rates  of  walking.  In 
Fig.  26,  AB  represents  5 J  hours,  B  C  3f 
hours,  and  AD  the  distance  required. 
A  H  shows  his  movement  during  the  first 
half  of  the  ascent,  H  E  that  during  the 
second  half,  and  E  C  his  descent. 

A  sets  off  from  London  to  York,  and 
B  at  the  same  time  from  York  to  Lon- 
don, and  they  travel  uniformly.  A 
reaches  York  16  hours  and  B  reaches 
London  36  hours  after  they  have  met  on 
the  road.     Find  in  what  time  each  has 


55 


performed  the  journey.     In  Fig.  27,  A  F 


represents  A's  movement,  and  D  C  that 


56 

of  B  ;  EF  is  sixteen  and  BC  36  hours. 
The  intersection  of  FA  with  AC  and 
of  C  D  with  the  lower  horizontal  must 
fall  on  the  same  vertical,  A  D. 

Two  trains  of  cars,  92  feet  and  84  feet 
long  respectively,  are  moving  with  uni- 
form velocities  on  parallel  tracks.  When 
they  go  in  opposite  directions  they  pass 
each  other  in  one  second  and  a  half ;  but 
when  they  go  in  the  same  direction  the 
faster  train  passes  the  other  in  six  sec- 
onds. Find  the  rate  at  which  each  train 
moves. 

Two  travelers,  A  and  B,  start  from 
two  places,  P  and  Q,  at  the  same  time. 
A  starts  from  P  with  the  design  to  pass 
through  Q,  and  B  starts  from  Q  and 
travels  in  the  same  direction  as  A.  When 
A  overtook  B  it  was  found  that  they  had 
together  traveled  thirty  miles,  that  A 
had  passed  through  Q  four' hours  before, 
and  that  B  at  his  rate  of  traveling  was 
nine  hours'  journey  distant  from  P. 
Find  the  distance  between  P  and  Q.  In 
Fig.  28,  PR  is  nine  hours,  T  S  is  four 
hours,  and  P  V  plus  Q  Y  is  thirty  miles. 


51 


Fig.  28. 

P  Q,  the  distance  required,  must  be  such 
that  Q  S  being  drawn  parallel  to  P  L, 
S  P  shall  cut  Q  M  in  a  point,  O,  which 
shall  be  four  hours  back  of  S. 

We  will  conclude  by  an  application  of 
the  graphic  method  to  a  question  of  great 
practical  importance,  viz.  the  adjustment 
of  the  running  times  of  railway  trains, 
which,  as  before  stated,  has  been  for  a 
long  time  employed  by  railway  mana- 
gers, and  which  first  suggested  to  the 
writer  the  solutions  given  in  the  preced- 
ing pages. 

Let  the  heavy  vertical  lines,  in  Fig  29,* 
represent  the  successive  hours  of  the 
day,  and  the  intermediate  finer  lines  the 

*  Frontispiece. 


58  , 

quarter  hours.  The  horizontal  lines  rep- 
resent the  several  stations  along  the 
road,  the  vertical  distances  between 
them  being  laid  off  by  scale  according 
to  the  actual  distances  in  miles.  Sup- 
pose that  we  wish  to  start  a  train  at  six 
o'clock  A.  M.,  from  the  station  represent- 
ed by  the  line  A  A,  so  that  it  shall  arrive 
at  the  station  shown  by  the  line  J  J  at 
three  o'clock  p.  m.  stopping  fifteen 
minutes  at  each  way  station.  The  num- 
ber of  way  stations  being  eight,  the 
whole  time  consumed  by  stops  will  be 
120  minutes,  or  two  hours.  From  3  p.  m. 
upon  the  lower  horizontal  line  we  go 
back  two  hours,  or  to  1  p.  m.  and  from 
6  A.  M.  upon  the  upper  horizontal  we 
draw  a  line  which  produced  would  strike 
1  p.  M.  upon  the  lower  line.  This  diagon- 
al reaches  the  line  BB  at  7:23  a.  m.  As 
we  stop  at  the  station  15  minutes,  we 
pass  along  on  the  line  B  B  a  distance 
equal  to  fifteen  minutes  on  the  time 
scale,  and  from  the  point  thus  reached 
we  start  again  parallel  to  the  first 
diagonal,  ariving  at  station  C  at  8:20  a. 


69 

if.  Proceeding  in  the  same  way  we 
arrive  at  station  J  at  3  p.  m.,  as  desired. 
The  inclination  of  the  diagonal  shows 
the  speed. 

If  we  would  start  a  train  from  station 
A  at  8:30  a.  m.  to  arrive  at  J  at  11:15, 
making  no  stops,  it  would  pass  the  train 
above  described  at  station  D,  and  will 
run  the  whole  distance  in  two  hours  and 
45  minutes.  Trains  running  in  the  oppos- 
ite direction  are  shown  on  the  diagram 
by  diagonals  ascending  from  left  to 
right.  Thus  a  train  leaving  station  J  at 
6  A.  M.,  to  arrive  at  station  A  at  noon 
making  no  stops,  will  run  as  by  the 
broken  diagonal  from  6  a.m.,  on  the  lower 
line  to  12  on  the  upper  one,  passing  the 
6  A.  M  and  the  8:30  a.  m.  trains,  run- 
ning in  the  opposite  direction,  at  station 
D.  It  will  be  observed  that  the  line 
from  6  to  12  changes  its  rate  of  inclina- 
tion at  the  horizontal  D,  by  which  we 
understand  that  the  train  changes  its  rate 
of  speed  at  that  station,  running  faster 
from  D  to  A  than  from  J  to  D. 

If  it  is  desired  to  work  a  construction 


60 

train  between  the  stations  E  and  D  from 
6  A.  M.  to  6  p.  M.,  the  movement  of 
such  a  train  is  shown  by  the  short  diag- 
onals between  the  horizontals  D  and  E, 
and  its  time  card  would  be  as  follows  : 
Leave  E  at  6  a.  m.,  and  arrive  at  D  at 
1.  Leave  D  at  7:15  and  arrive  at  E  at 
8:15.  Leave  E  at  8:30  and  arrive  at  D 
at  9:30,  crossing  the  6  a.  m.  and  the 
8:30  A.  M.  trains  from  A  to  J,  and  being 
passed  by  the  6  a.  m.  train  from  J  to  A. 
Leave  D  at  10  and  arrive  at  E  at  11. 
Leave  E  at  11:15  and  arrive  at  D  at  12:- 
15,  and  wait  to  be  passed  by  the  9  a.  m. 
train  from  J  to  A.  Leave  D  at  12:45 
p.  M.  and  arive  at  E  at  1 :30  P.  M.,  leave 
E  at  1:45  and  arrive  at  D  at  2:45,  and 
pass  noon  train  from  station  A,  and  11:15 
A.  M.  train  from  station  J.  Leave  D  at 
3:15  and  arrive  at  E  at  4  p.  m.  Leave  E  at 
4:15  and  arrive  at  D  at  5  p.  m.  Leave 
D  at  5:15  and  arrive  at  E  at  6  P.  M. 

If  a  train  leaves  A  at  noon  and  runs 
towards  J,  leaving  C  at  2:05  and  reach- 
ing E  at  3:20,  and  another  train  leaves 
J  at  11:15  A.  M.,  and  G^  at    1  p.  m.,   run- 


61 

ning  to  A  as  by  the  diagonal,  without 
stopping,  the  trains  will  pass  at  3  p.  m. 
at  a  point  between  D  and  E,the  exact  posi- 
tion of  which  may  be  found  by  the  scale 
of  miles  according  to  which  the  length  of 
the  road  or  the  distance  A  J  is  plotted, 
at  which  place  ,a  side  track  must  be  pro- 
vided. 

In  practice  the  diagram  is  accurately 
drawn  to  a  large  scale,  and  the  several 
trains  are  represented  by  differently 
colored  elastic  lines  fastened  by  pins  so 
that  they  may  be  moved  from  hour  to 
hour  through  the  day  and  night  as  the 
various  occurences  upon  the  road  may  de- 
mand, some  trains  being  hastened  others 
retarded,  extras  put  in  and  all  provisions 
made  for  securing  regularity  in  the 
movement  and  freedom  from  disaster. 
The  grades  and  curves  may  if  desirable 
be  shown  upon  the  vertical  line  A  J,  by 
which  those  parts  of  the  road  may  at 
once  be  seen  where  from  increased  resist- 
ance a  lower  speed  will  need  to  be  adopt- 
ed.    Upon  a  double  track  road  a  chart 


62 


may  be  prepared  for  each  track,  and 
diagonals  in  one  direction  only  will 
appear  upon  each  diagram. 


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plates,  together  with  a  Travel  Scale,  and  numerous 
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VAN  BUREN.  Investigations  of  Formulas,  for  the 
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JOYNSON.     Designing  and  Construction  of  Machine 

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GILLMORE.  Coignet  Beton  and  other  Artificial  Stone. 
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SAELTZER.  Treattse  on  Acoustics  in  connection  with 
Ventilation.  By  Alexander  Saeltzer,  Architect. 
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THE  EARTH'S  CRUST.  A  handy  Outline  of  Geo- 
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DICTIONARY  of  Manufactures,  Mining,  Machinery, 
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BOW.  A  Treatise  on  Bracing,  with  its  application  to 
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3 


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GILLMORE  (Gen.  Q.  A.)  Treatise  on  Limes,  Hy- 
draulic Cements,  and  Mortars.  Papers  on  Practical 
Engineering,  U.  S.  Engineer  Department,  No.  9, 
containing  Reports  of  numerous  Experiments  con- 
ducted in  New  York  City,  during  the  years  1858  to 
1 86 1,  inclusive.  By  Q.  A.  Gillmore,  Bvt.  Maj  -Gen., 
U.  S.  A.,  Major,  Corps  of  Engineers.  With  num- 
erous illustrations,     i  vol,  8vo,  cloth $4  00 

HARRISON.  The  Mechanic's  Tool  Book,  with  Prac- 
tical Rules  and  Suggestions  for  Use  of  Machinists, 
Iron  Workers,  and  others.  By  W.  B.  Harrison, 
associate  editor  of  the  "American  Artisan."  Illus- 
trated with  44  engravings.     i2mo,  cloth 150 

HENRICI  (Olaus).  Skeleton  Structures,  especially  in 
their  application  to  the  Building  of  Steel  and  Iron 
Bridges.  By  Olaus  Henrici.  With  folding  plates 
and  diagrams,     i  vol.  8vo,  cloth. i  50 

HEWSON  ( Wm.)  Principles  and  Practice  of  Embank 
ing  Lands  from  River  Floods,  as  applied  to  the  Le- 
)  vees  of  the  Mississippi.  By  William  Hewson,  Civil 
Engineer,     i  vol.  8vo,  cloth 200 

HOLLEY  (A.  L.)  Railway  Practice.  American  and 
European  Railway  Practice,  in  the  economical  Gen- 
eration of  Steam,  including  the  Materials  and  Con- 
struction of  Coal-bumin§  Boilers,  Combustion,  the 
Variable  Blast,  Vaporization,  Circulation,  Superheat- 
ing, Supplying  and  Heating  Feed-water,  etc.,  and 
the  Adaptation  of  Wood  and  Coke-burning  Engines 
to  Coal-burning ;  and  in  Permanent  Way,  including 
Road-bed,  Sleepers,  Rails,  Joint-fastemngs,  Street 
Railways,  etc.,  etc.  By  Alexander  L.  Holley,  B.  P. 
With  77  lithographed  plates,     i  vol.  folio,  cloth 12  00 

KING  (W.  H.)    Lessons  and  Practical  Notes  on  Steam, 

the  Steam  Engine,  Propellers,  etc.,  etc.,  for  Young 

Marine   Engineers,  Students,   and   others.     By  the 

^    late  W.  H.  King,  U.  S.  Navy.     Revised  by  Chief 

"    Engineer  J.  W.  King,  U.  S.  Navy.    Twelfth  edition, 

enlarged.    8vo,  cloth 2  00 

MINIFIE  (Wm.)  Mechanical  Drawing.  A  Text-Book 
of  Geometrical  Drawing  for  the  use  of  Mechsmic* 

4 


Z,.   VAN   NOSTBANDS   PUBLICATIONS. 

an&  Schools,  in  which  the  Definitions  and  Rules  ol 
Geometry  are  familiarly  explained;  the  Practical 
Problems  are  arranged,  from  the  most  simple  to  the 
more  complex,  and  m  their  description  technicalities 
are  avoided  as  much  as  possible.  With  illustrations 
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ways and  Machinery ;  an  Introduction  to  Isometrical 
Drawing,  and  an  Essay  on  Linear  Perspective  and 
Shadows.  Illustrated  with  over  200  diagrams  en- 
graved on  steel.  By  Wm.  Minifie,  Architect.  Sev- 
enth edition.    With  an  Appendix  on  the  Theory  and 

Application  of  Colors,     i  vol.  8vo,  cloth $4  00 

"It  Is  the  best  work  on  Drawing  that  we  have  ever  seen,  and  Is 
eapecially  a  text-book  of  Geometrical  Drawing  lor  the  use  of  Mechanics 
and  Schools.  No  young  Mechanic,  such  as  a  Machinists,  Engineer,  Cabi- 
net-maker, Millwright,  or  Carpenter,  should  be  without  iW— Scientific 
American. 

Geometrical  Drawing.    Abridged  from  the  octavo 

edition,  for  the  use  of  Schools.  Illustrated  with  48 
steel  plates.     Fifth  edition,     i  vol.  i2mo,      cloth...,  20c 

STILLMAN  (Paul.)  Steam  Engine  Indicator,  and  the 
Improved  Manometer  Steam  and  Vacuum  Gauges — 
their  Utility  and  Application.  By  Paul  Stillman. 
New  edition,     i  vol.  i2mo,  flexible  cloth i  00 

SWEET  (S.H.)  Special  Report  on  Coal ;  showing  its 
Distribution,  Classification,  and  cost  delivered  over 
different  routes  to  various  points  in  the  State  of  New 
York,  and  the  principal  cities  on  the  Atlantic  Coast. 
By  S.  H.  Sweet.    With  maps,  i  vol.  Svo,  cloth 3  00 

WALKER  (W.  H.)  Screw  Propulsion.  Notes  on 
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W.  H.  Walker,  U,  S.  Navy,     i  vol.  Svo,  cloth 75 

WARD  (J.  H.)  Steam  for  the  Million.  A  popular 
Treatise  on  Steam  and  its  Application  to  the  Useful 
Arts,  especially  to  Navigation.  By  J.  H.  Wand, 
Commander  U.  S.  Navy.  New  and  revised  edition. 
I  vol.  Svo,  cloth 1  00 

WEISBACH  (Julius).     Principles  of  the  Mechanics  of     • 
Machinery  and  Engineering.     By  Dr.  Julius  Wels- 
^   bach,  of  Freiburg.    Translated  fi-om  the  last  German 

edition.    '  Vol.  I.,  Svo,  cloth 1000 

5 


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DIEDRICH.  The  Theory  of  Strains,  a  Compendium 
for  the  calculation  and  construction  of  Bridges,  Roofs, 
and  Cranes,  with  the  application  of  Trigonometrical 
Notes,  containing  the  most  comprehensive  informa- 
tioti  in  regard  to  the  Resulting  strains  for  a  perman- 
ent Load,  as  also  for  a  combined  (Permanent  and 
Rolling)  1-oad.  In  two  sections,  adadted  to  the  re- 
quirements of  the  present  time.  By  John  Diedrich, 
0.  E.     Illustrated  by  numerous  plates  and  diagrams, 

8vo,  cloth * (t****.*.     5  oo 

WILLIAMSON  (R.  S.)  On  the  use  of  the  Barometer  on 
Surveys  and  Reconnoissances.  Part  I.  Meteorology 
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metric Hypsometry.  By  R.  S.  Wiliamson,  Bvt 
Lieut.-Col.  U.  S.  A.,  Major  Corps  of  Engineers. 
With  Illustrative  Tables  and  Engravings.  Paper 
No.   15,  Professional  Papers,  Corps   of  Engineers. 

I  vol.  4to,  cloth 15  00 

POOK  (S.  M.)  Method  of  Comparing  the  Lines  and 
Draughting  Vessels  Propelled  by  Sail  or  Steam. 
Including  a  chapter  on  Laying  off  on  the  Mould- 
Loft  Floor.  By  Samuel  M.  Pook,  Naval  Construc- 
tor.    I  vol.  8vo,  with  illustrations,  cloth 5  00 

ALEXANDER  (J.  H.)  Universal  Dictionary  of 
Weights  and  Measures,  Ancient  and  Modem,  re- 
duced to  the  standards  of  the  United  States  of  Ame- 
rica.    By  J.  H.  Alexander.     New  edition,  enlarged. 

1  vol.  8vo,  cloth 3  50 

WANKLYN.  A  Practical  Treatise  on  the  Examination 
of  Milk,  and  its  Derivatives,  Cream,  Butter  and 
Cheese.     By  J.  Alfred  Wanklyn,  M.  R.  C.  S.,  i2mo, 

cloth 1  00 

RICHARDS'  INDICATOR.  A  Treatise  on  the  Rich- 
ards Steam  Engine  Indicator,  with  an  Appendix  by 
F.  W.  Bacon,  M.  E.     i8mo,  flexible,  cloth x  •• 

6 


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POPE  Modern  Practice  of  the  Electric  Telegraph.  A 
Hand  Book  for  Electricians  and  operators.  By  Frank 
L,  Pope  Eighth  edition,  revised  and  enlarged,  and 
fully  ililustrated.     8vo,  cloth $2.00 

"  There  is  no  other  work  of  this  kind  In  the  English  langnage  that  con- 
tains in  so  small  a  compaaflso  much  practical  information  in  the  appli- 
ntion  of  galvanic  electricity  to  telegraphy.  It  should  be  in  the  hnndaof 
erery  cue  interested  in  telegraphy,  or  the  use  of  Batteries  for  oth«r  pur* 
puses.' 

MORSE.  Examination  of  the  Telegraphic  Apparatus 
and  the  Processes  in  Telegraphy.  By  Samuel  F. 
Morse,  LL.D.,  U.  S.  Commissioner  Paris  Universal 
Exposition,  1867.     Illustrated,  Svo,  cloth $2  00 

SABINE.  History  and  Progress  of  the  Electric  Tele- 
graph, with  descriptions  of  some  of  the  apparatus. 
By  Robert  Sabine,  C.  E.  Second  edition,  with  ad- 
ditions, i2mo,  cloth I  35 

BLAKE.  Ceramic  Art.  A  Report  on  Pottery,  Porce- 
lain, Tiles,  Terra  Cotta  and  Brick.  By  W.  P.  Blake, 
U.  S.  Commissioner,  Vienna  Exhibition,  1873.  8vo, 
cloth 2  00 

BENET.  Electro-Ballistic  Machines,  and  the  Schultz 
Chronoscope.  By  Lieut -Col.  8.  V.  Benet,  Captain 
of  Ordnance.  U.  S.  Army.  Illustrated,  second  edi- 
tion, 4to,  cloth 3  00 

MICHAELIS.  The  Le  Boulenge  Chronograph,  with 
three  Lithograph  folding  plates  of  illustrations.  By 
Brevet  Captain  O.  E.  Michaelis,  First  Lieutenant 
Ordnance  Corps,  U.  S .  Army,     4to,  cloth 3  00 

ENGINEERING  FACTS  AND  FIGURES  An 
Annual  Register  of  Progress  in  Mechanical  Engineer- 
ing and  Construction,  for  the  years  1863,  64,  65,  66, 
67,  68.  Fully  illustrated,  6  vols.  i8mo,  cloth,  $2.50 
per  vol.,  each  volume  sold  separately 

HAMILTON.  Useful  Information  for  Railway  Men. 
Compiled  by  W.  G.  Hamilton,  Engineer.  Fifth  edi- 
tion, revised  and  enlarged,  562  pages  Pocket  form. 
Morocco,  gilt 2  00 

7 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 


STUART.  The  Civil  and  Military  Engineers  of  Amer- 
ica. By  Gen.  C.  B.  Stuart.  With  9  finely  executed 
portraits  of  eminent  engineers,  and  illustrated  by 
engravings  of  some  of  the  most  important  works  con- 
structed m  America.     8vo,  cloth $5  00 

STONE Y.  The  Theory  of  Strains  in  Girders  and  simi- 
lar structures,  with  observations  on  the  application  of 
Theory  to  Practice,  and  Tables  of  Strength  and  other 
properties  of  Materials.  By  Bindon  B.  Stoney,  B.  A. 
New  and  revised  edition,  enlarged,  with  numerous 
engravings  on  wood,  by  Oldham.  Royal  8vo,  664 
pages.     Complete  in  one  volume.     8vo,  cloth 12  50 

SHRE VE.  A  Treatise  on  the  Strength  of  Bridges  and 
Roofs.  Comprising  the  determination  of  Algebraic 
formulas  for  strains  in  Horizontal,  Inclined  or  Rafter, 
Triangular,  Bowstring,  Lenticular  and  other  Trusses, 
from  fixed  and  ^noving  loads,  with  practical  applica- 
tions and  examples,  for  the  use  of  Students  and  Engi- 
neers. By  Samuel  H.  Shreve,  A.  M. ,  Civil  Engineer. 
87  wood  cut  illustrations.    8vo,  cloth 5  00 

MERRILL.  Iron  Truss  Bridges  for  Railroads.  The 
method  of  calculating  strains  in  Trusses,  with  a  care- 
ful comparison  of  the  most  prominent  Trusses,  in 
reference  to  economy  in  combination,  etc,  etc  By 
Brevet.  Col.  William  E.  Merrill,  U  S.  A.,  Major 
Corps  of  Engineers,  with  nine  lithographed  plates  of 
Illustrations.     4to,  cloth 5  00 

WHIPPLE.  An  Elementary  and  Practical  Treatise  on 
Bridge  Building.  An  enlarged  and  improved  edition 
of  the  author's  original  work.  By  S.  Whipple,  C.  E. , 
inventor  of  the  Whipple  Bridges,  &c.  Illustrated 
8vo,  cloth 4  00 

THE  KANSAS  CITY  BRIDGE.  With  an  account 
of  the  Regimen  of  the  Missouri  River,  and  a  descrip- 
tion of  the  methods  used  for  F'ounding  in  that  River. 
ByO.  Chanute,  Chief  Engineer,  and  George  Morri- 
son, Assistant  Engineer.  Illustrated  with  five  litho- 
graphic views  and  twelve  plates  of  plans.  4to,  cloth,  6  00 
8 


D.  y^^:^  NOSTKAND  S  rUBWCATIONS. 


MAC  CORD.  A  Practical  Treatise  on  the  Slide  Valve 
by  Eccentrics,  examining  by  methods  the  action  of  the 
Eccentric  upon  the  Slide  Valve,  and  explaining  the 
Practical  processes  of  laying  out  the  movements, 
adapting  the  valve  for  its  various  duties  in  the  steam 
engme.  For  the  use  of  Kngineers,  Draughtsmen, 
Machinists,  and  Students  of  Valve  Motions  in  gene- 
ral. By  C.  W.  Mac  Cord,  A.  M. ,  Professor  ofMe- 
chanical  Drawing,  Stevens*  Institute  of  Technology, 
Hoboken,  N.  J.  Illustrated  by  8  full  page  copper- 
plates.    4to,  cloth $4  oo 

KIRK  WOOD.  Report  on  the  Filtration  of  River 
Waters,  for  the  supply  of  cities,  as  practised  in 
Europe,  made  to  the  Board  of  Water  Commissioners 
of  the  City  of  St.  Louis.  By  James  P.  Kirkwood. 
Illustrated  by  30  double  plate  engravings.    4to,  cloth,  15  00 

PLATTNER.  Manual  of  Qualitative  and  Quantitative 
Analysis  with  the  Blow  Pipe.  From  the  last  German 
edition,  revised  and  enlarged.  By  Prof.  Th.  Richter, 
of  the  Royal  Saxon  Mining  Academy.  Translated 
by  Prof  H.  B.  Cornwall,  Assistant  in  the  Columbia 
School  of  Mines,  New  York  assisted  by  John  H. 
Caswell.  Illustrated  with  87  wood  cuts,  and  one 
lithographic  plate.  Second  edition,  revised,  560 
pages,  8vo,  cloth 7  50 

PLYMPTON.  The  Blow  Pipe.  A  Guide  to  its  Use 
in  the  Determination  of  Salts  and  Minerals.  Com- 
piled from  various  sources,  by  George  W.  Plympton, 
C  E.  A.  M.,  Professor  of  Physical  Science  in  the 
Polytechnic  Institute,  Brooklyn,  New  York,  izmo, 
cloth I  so 

PYNCHON.  Introduction  to  Chemical  Physics,  design- 
ed for  the  use  of  Academies,  Colleges  and  High 
Schools.  Illustrated  with  numerous  engravings, and 
containing  copious  experiments   with  directions  for 

Preparing  them.  By  Thomas  Ruggles  Pynchon, 
I.  A.,  Professor  of  Chemistry  and  the  Natural  Sci- 
ences, Trinity  College,  Hartford  New  edition,  re- 
vised and  enlarged  and  illustrated  by  269  illustrations 
on  wood.     Crown,  8vo.  cloth 3  o» 

9 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 

ELIOT  AND  STORER.  A  compendious  Manual  of 
Qualitative  (Chemical  Analysis.  By  Charles  W. 
Eliot  and  Frank  H.  Storer.  Revised  with  the  Co- 
operation of  the  authors.  By  William  R.  Nichols, 
Professor  of  Chemistry  in  the  Massachusetts  Insti- 
tute of  Technology      Illustrated,  i2mo,  cloth $i  50 

RAM  M  ELS  BERG.  Guide  to  a  course  of  Quantitative 
Chemical  Analysis,  especially  of  Minerals  and  Fur- 
nace Products.  Illustrated  by  Examples  By  C.  F. 
Ramn?  alsberg.  Translated  by  J.  Towler,  M.  D. 
8 vo,  cloth 2  35 

EGLESTON.  Lectures  on  Descriptive  Mineralogy,  de- 
livered at  the  School  of  Mines,  Columbia  College. 
By  Professor  T.  Egleston.  Illustrated  by  34  Litho- 
graphic Plates.     8vo,  cloth 450 

JACOB.  On  the  Designing  and  Construction  of  Storage 
Reservoirs,  with  Tables  and  Wood  Cuts  representing 
Sections,  &c.,  iSmo,  boards 50 

WATT'S  Dictionary  of  Chemistry      New  and  Revised     ^ 
edition  complete  in  6  vols.   Svo  cloth,  $62.00     Sup- 
plementary volume  sold  separately.     Price,  cloth. ..     900 

RANDALL.  Quartz  Operators  Hand- Book.  By  P.  M. 
Randall.  New  edition,  revised  and  enlarged,  fully 
illustrated.     i2mo»  clotb   200 

SILVERSMITH.  A  Practical  Hand-Book  for  Miners, 
Metallurgists,  and  Assayers,  comprising  the  most  re- 
cent improvements  in  the  disintegration,  amalgama- 
tion, smelting,  and  parting  of  the  Frecious  ores,  with 
a  comprehensive  Digest  of  the  Mining  Laws.  Greatly 
augmented,  revised  and  corrected.  By  Julius  Silver- 
smith. Fourth  edition.  Profusely  illustrated.  i2mo, 
cloth • 3  00 

THE  USEFUL  METALS  AND  THEIR  ALLOYS. 
including  Mining  Ventilation,  Mining  Jurisprudence, 
and  Metallurgic  Chemistry  employed  in  the  conver- 
sion of  Iron,  Copper,  Tin,  Zinc,  Antimony  and  Lead 
ores,  with  their  applications  to  the  Industrial  Arts. 
By  Scoflfren,  Truan,  Clay,  Oxland,  Fairbaim,  and 

others    Fifth  edition,  half  calf , 375 

JO 


D.  VAN  NOSTBAND  S  PUBLICATIONS. 

JOYNSON.  The  Metals  used  in  construction,  Iron, 
Steel,  Bessemer  Metal,  etc.,  etc.  By  F.  H.  Joynson. 
Illustrated,  i2mo,  cloth $o  71 

VON  COTTA.  Treatise  on  Ore  Deposits.  By  Bern- 
hard  Von  Cotta,  Professor  of  Geology  in  the  Royal 
School  of  Mines,  Freidberg,  Saxony.  Translated 
from  the  second  German  edition,  by  Frederick 
Prime,  J  r..  Mining  Engineer,  and  revised  by  the  au- 
thor, with  numerous  illustrations.     8vo,  cloth 4  00 

GREENE.  Graphical  Method  for  the  Analysis  of  Bridge 
Trusses,  extended  to  continuous  Girders  and  Draw 
Spans.  By  0.  K.  Greene,  A.  M.,  Prof,  of  Civil  Engi- 
neering, University  of  Michigan.  Illustrated  by  3 
folding  plates,  8vo,  cloth 2  00 

BELL.  Chemical  Phenomena  of  Iron  Smelting.  An 
experimental  and  practical  examination  of  the  cir- 
cumstances which  determine  the  capacity  of  the  Blast 
Furnace,  The  Temperature  of  the  air,  and  the 
proper  condition  of  the  Materials  to  be  operated 
upon.    By  I.  Lowthian  Bell.    8 vo,  cloth 600 

ROGERS.  The  Geology  of  Pennsylvania.  A  Govern- 
ment survey,  with  a  general  view  of  the  Geology  of 
the  United  States,  Essays  on  the  Coal  Formation  and 
its  Fossils,  and  a  description  of  the  Coal  Fields  of 
North  America  and  Great  Britain.  By  Henry  Dar- 
win Rogers,  late  State  Geologist  of  Pennsylvania, 
Splendidly  illustrated  with  Plates  and  Engravings  in 
the  text.     3  vols.,  4to,  cloth,  with  Portfolio  of  Maps.  30  00 

BURGH,  Modern  Marine  Engineering,  applied  to 
Paddle  and  Screw  Propulsion.  Consisting  of  36 
colored  plates,  259  Practical  Wood  Cut  Illustrations, 
and  403  pages  ot  descriptive  matter,  the  whole  being 
an  exposition  of  the  present  practice  of  James 
Watt  &  Co.,  J.  &  G.  Rennie,  R.  Napier  &  Sons, 
and  other  celebrated  firms,  by  N.  P.  Burgh,  Engi- 
neer, thick  4to,  vol.,  doth,  $25.00 ;  half  mor 30  00 

CHURCH.    Notes  of  a  Metallurgical  Journey  in  Europe. 

By  J.  A.  Church,  Engineer  of  Mines,  Svo,  cloth s  co 

11 


D.  VAN  NOSTRAND'S  PUBLICATIONS, 

BOCJRNE.  Treatise  on  the  Steam  Engine  in  its  various 
applications  to  Mines,  Mills,  Steam  Navigation^ 
Railways,  and  Agriculture,  with  the  theoretical  in- 
vestigations respecting  the  Motive  Power  of  Heat, 
and  the  proper  proportions  of  steam  engines.  Elabo- 
rate tables  of  the  right  dimensions  of  every  part,  and 
Practical  Instructions  for  the  manufacture  and  man- 
agement of  every  species  of  Engine  in  actual  use. 
By  John  Bourne,  being  the  ninth  edition  of  "  A 
Treatise  on  the  Steam  Engine,"  by  the  "  Artizan 
Club«"  Illustrated  by  38  plates  and  546  wood  cuts. 
4to,  cloth „ $15  00 

STUART.  The  Naval  Dry  Docks  of  the  United 
Sjates.  By  Charles  B.  Stuart  late  Engineer-in-Chief 
of  the  U.  S.  Navy.  Illustrated  with  24  engravings 
on  steel.     Fourth  edition,  cloth 600 

ATKINSON.     Practical  Treatises  on  the   Gases  met 

with  in  Coal  Mines.     i8mo,  boards 50 

FOSTER.  Submarine  Blasting  in  Boston  Harbor, 
Massachusetts.  Removal  of  Tower  and  Corwin 
Rocks.  By  J.  G.  Foster,  Lieut -Col.  of  Engineers, 
U.  S . .  Army.  Illustrated  with  seven  plates,  4to, 
cloth 3  50 

BARNES  Submarine  Warfare,  offensive  and  defensive, 
including  a  discussion  of  the  offensive  Torpedo  Sys- 
tem, its  effects  upon  Iron  Clad  Ship  Systems  and  m- 
fluence  upon  future  naval  wars.  By  Lieut. -Com- 
mander J.  S.  Barnes,  U.  S.  N.,  with  twenty  litho- 
graphic plates  and  many  wood  cuts.     8vo,  cloth. . .    .     5  00 

HOLLEY.  A  Treatise  on  Ordnance  and  Armor,  em- 
bracing descriptions,  discussions,  and  professional 
opinions  concerning  the  materials,  fabrication,  re- 
quirements, capabilities,  and  endurance  of  European 
and  American  Guns,  for  Is  aval,  Sea  Coast,  and  Iron 
Clad  Warfare,  and  their  Rifling,  Projectiles,  and 
Breech- Loading ;  also,  results  of  experiments  against 
armor,  from  official  records,  with  an  appendix  refer- 
ring to  Gun  Cotton,  Hooped  Guns,  etc.,  etc  By 
Alexander  L.  Holley,  B.  P.,  948  pages,  493  engrav- 
ings, and  J  47  Tables  of  Resuhs,  etc,  8vo,  half  roan.  10  00 
12 


D.  VAJS  NOSTKAND'S  PUBLICATIONS. 

SIMMS.  A  Treatise  on  the  Principles  and  Practice  of 
Levelling,  showing  its  application  to  purposes  of 
Railway  Engineering  and  the  Construction  of  Roads, 
&C.  By  Frederick  W.  Simms,  C.  E.  From  the  sth 
London  edition,  revised  and  corrected,  with  the  addf- 
tion  of  Mr.  Laws's  Practical  Examples  for  setting 
out  Railway  Curves.  Illustrated  with  three  Litho- 
graphic plates  and  numerous  wood  cuts.     8vo,  cloth.  $2  50 

BURT.  Key  to  the  Solar  Compass,  and  Surveyor's 
Companion ;  comprising  all  the  rules  necessary  for 
use  in  the  field ;  also  description  of  the  Linear  Sur- 
reys and  Public  Land  System  of  the  United  States, 
Notes  on  the  Barometer,  suggestions  for  an  outfit  for 
a  survey  of  four  months,  etc  By  W.  A.  Burt,  U.  S. 
Deputy  Surveyor.  Second  edition.  Pocket  book 
form,  tuck 2  50 

THE  PLANE  TABLE.  Its  uses  in  Topographical 
Surveying,  from  the  Papers  of  the  U.  S.  Coast  Sur- 
vey.    Illustrated,  8 vo,  cloth 2  ©o 

"  This  worK  gives  a  description  of  the  Plane  Table,  employed  at  th« 

U.  S.  Coast  Purvey  ofBce,  and  the  manner  of  using  it." 

JEFFER'S.  Nautical  Surveying.  By  W.  N.  Jeffers, 
Captain  U.  S.  Navy.  Illustrated  with  9  copperplates 
and  31  wood  cut  iHustrations.     8vo,  cloth 5  cxa 

CH  AUVENET.  New  method  of  correcting  Lunar  Dis- 
tances, and  improved  method  of  Finding  the  error 
and  rate  of  a  chronometer,  by  equal  altitudes.  By 
W.  Chauvenet,  LL.D.    8vo,  cloth 2  00 

BRUNNOW.  Spherical  Astronomy.  By  F.  Brunnow, 
Ph.  Dr.  Translated  by  the  author  from  the  second 
German  edition.      8vo,  cloth 6  jo 

PEIRCE.  System  of  Analytic  Mechanics.  By  Ben- 
jamin Peirce.     4to,  cloth 1000 

COFFIN.     Navigation  and  Nautical  Astronomy.     Pre- 

Pared  for  the  use  of  the  U.  S.  Naval  Academy.     By 
rof.  J.  H.  C.  Coffin.  Fifth  edition.  52  wood  cut  illus- 
trations.    z2mo,  cloth 3  50 

13 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 


CLARK.  Theoretical  Navigation  and  Nautical  Astron- 
omy. By  Lieut.  Lewis  Clark,  U.  S.  N.  Illustrated 
with  41  wood  cuts.     8vo,  cloth $3  00 

HASKlNS.  The  Galvanometer  and  its  Uses.  A  Man- 
ual for  Electricians  and  Students.  By  C  H.  Has- 
kins.     i2mo,  pocket  form,  morocco.    (In  press) 

GOUGE.  New  System  of  Ventilation,  which  has  been 
thoroughly  tested,  under  the  patronage  of  many  dis- 
tinguished persons.  By  Henry  A.  Gouge.  With 
many  illustrations.    8vo,  cloth 2  00 

BECKWITH.  Observations  on  the  Materials  and 
Manufacture  of  Terra-Cotta,  Stone  Ware,  Fire  Brick, 
Porcelain  and  Encaustic  Tiles,  with  remarks  on  the 
products  exhibited  at  the  London  International  Exhi- 
bition, 187 1.  By  Arthur  Beckwith,  C  E.  8vo, 
paper 60 

MORFIT.  A  Practical  Treatise  on  Pure  Fertilizers,  and 
the  chemical  conversipn  of  Rock  Guano,  Marlstones, 
Coprolites,  and  the  Crude  Phosphates  of  Lime  and 
Alumina  generally,  into  various  valuable  products. 
By  Campbell  Morfit,  M.D.,  with  28  illustrative  plates, 
8vo,  cloth 20  3o 

BARNARD.  Tne  Metric  System  of  Weights  and 
Measures.  An  address  delivered  before  the  convoca- 
tion of  the  University  of  the  State  of  New  York,  at 
Albany,  August,  1871.  By  F.  A.  P.  Barnard,  LL.D., 
President  of  Columbia  College,  New  York.  Second 
edition  from  the  revised  edition,  printed  for  the  Trus- 
tees of  Columbia  College.     Tinted  paper,  8vo,  cloth    3  00 

— Report  on  Machinery  and  Processes  on  the  In- 
dustrial Arts  and  Apparatus  of  the  Exact  Sciences, 
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position, 1867.     Illustrated,  8vo,  cloth •    5  00 

txLLAN.  Theorjr  of  Arches.  By  Prof.  W.  Allan,  for- 
merly of  Washington  &  Lee  University,  i8mo,  b'rds        50 

14 


D.  VAN  NOSTRAND  8  PUBLICATIONS. 


MYER.  Manual  of  Signals,  for  the  use  of  Signal  officers 
in  the  Field,  and  for  Military  and  Naval  Students, 
Military  Schools,  etc.  A  new  edition  enlarged  and 
illustrated.  By  Brig.  General  Albert  J.  Myer,  Chief 
Signal  Officer  of  the  army.  Colonel  of  the  Signal 
Corps  during  the  War  of  the  Rebellion.  i2mo,  48 
plates,  full  Roan $5  00 

WILLIAMSON.  Practical  Tables  in  Meteorology  and 
Hypsometry,  in  connection  with  the  use  of  the  Bar- 
ometer. By  CoL  R.  S.  Williamson,  U.  S.  A.  4to, 
cloth « 2  50 

CLEVENGER.  A  Treatise  on  the  Method  of  Govern- 
ment Surveying,  as  prescribed  by  the  U.  S.  Congress 
and  Commissioner  of  the  General  Land  Office,  with 
complete  Mathematical,  Astronomical  and  Practical 
Instructions  for  the  Use  of  the  United  States  Sur- 
veyors in  the  Field.  By  S.  R.  Oevenger,  Pocket 
Book  Form,  Morocco 2  50 

PICKERT  AND  METCALF.  The  Art  of  Graining. 
How  Acquired  and  How  Produced,  with  description 
of  colors,  and  their  application.  By  Charles  Pickert 
and  Abraham  Metcalf.  Beautifully  illustrated  with 
42  tinted  plates  of  the  various  woods  used  in  interior 
finishing.    Tinted  paper,  4to,  cloth 10  00 

HUNT.  Designs  for  the  Gateways  of  the  Southern  En- 
trances to  the  Central  Park.  By  Richard  M.  Hunt. 
With  a  description  of  the  designs.    4to.  cloth 5  00 

LAZELLE.  One  Law  in  Nature.  By  Capt.  H.  M. 
Lazelle,  U.  S.  A.  A  new  Corpuscular  Theory,  com- 
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its  Multiple  Atom  Constitution,  applied  to  the  Physi- 
cal Affections  or  Modes  of  Energy.     i2mo,  cloth. . .     i  50 

CORFIELD.  Water  and  Water  Supply.  By  W.  H. 
Corfield,  M.  A.  M,  D.,  Professor  of  Hygiene  and 
PubHc- Health  at  University  College,  London.  i8mo, 
boards 50 

15 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 


BOYNTON.  History  of  West  Point,  its  Military  Im- 
portance during  the  American  Revolution,  and  the 
Origin  and  History  of  the  U.  S.  Military  Academy. 
Bjr  Bvt  Major  C.  E.  Boynton,  A.M.,  Adjutant  of  the 
Military  Academy.  Second  edition,  416  pp.  8vo, 
printed  on  tinted  paper,  beautifully  illustrated  with 
36  maps  and  fine  engravings,  chiefly  from  photo- 
graphs taken  on  the  spot  by  the  author.  Extra 
cloth » .^ $3  S® 

WOOD.  West  Point  Scrap  Book,  being  a  collection  of 
Legends,  Stories,  Songs,  etc,  of  the  U.  S.  Military 
Academy.  By  Lieut.  O.  E.  Wood,  U.  S.  A.  Illus- 
trated by  69  engravings  and  a  copperplate  map. 
Beautifully  printed  on  tmted  paper.     8vo,  cloth 5  00 

WEST  POINT  LIFE.  A  Poem  read  before  the  Dia- 
lectic Society  of  the  United  States  Military  Academy. 
Illustrated  with  Pen-and-ink  Sketches.  By  a  Cadet. 
To  which  is  added  the  song,  "  Benny  Havens,  oh  1" 
oblong  8vo,  21  fiill  page  illustrations,  cloth 2  50 

GUIDE  TO  WEST  POINT  and  the  U.  S.  Military 
Academy,  with  maps  and  engravings,  iSmo,  blue 
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HENRY.  Military  Record  of  Civilian  Appointments  in 
the  United  States  Army.  By  Guy  V.  Henry,  Brevet 
Colonel  and  Captain  First  United  States  Artillery, 
Late  Colonel  and  Brevet  Brigadier  General,  United 
States  Volunteers.  Vol.  i  now  ready.  Vol.  2  in 
press.     Svo,  per  volume,  cloth 5  00 

HAMERSLY.  Records  of  Living  Officers  of  the  U. 
S.  Navy  and  Marine  Corps.  Compiled  from  official 
sources.  By  Lewis  B.  Hamersly,  late  Lieutenant 
U.  S.  Marine  Corps.     Revised  edition,  Svo,  cloth...     5  00 

MOORE.  Portrait  Gallery  of  the  War.  Civil,  Military 
and  Naval.  A  Biographical  record,  edited  by  Frank 
Moore.      60  fine  portraits  on  steel.     Royal  Svo, 

cloth 6  00 

16 


D   VAN  NOSTK.AND  8  PUBLICATIONS. 


PRESCOTT.  Outlines  of  Proximate  Organic  Analysis, 
for  the  Identification,  Separation,  and  Quantitative 
Determination  of  the  more  commonly  occurring  Or. 
ganic  Compounds.  By  Albert  B.  Prescott,  Professor 
of  Chemistry,  University  of  Michigan,  i2mo,  cloth...     i  75 

PRESCOTT.  Chemical  Examination  of  Alcoholic  Li- 
quors A  Manual  of  the  Constituents  of  the  Distilled 
Spirits  and  Fermented  Liquors  of  Commerce,  and 
their  Qualitative  and  Quantitative  Determinations. 
By  Albert  B.  Prescott,  i2mo,  cloth i  50 

BATTERSHALL.  Legal  Chemistry.  A  Guide  to  the 
Detection  of  Poisons,  Falsification  of  Writings,  Adul- 
teration of  Alimentary  and  Pharmaceutical  Substan- 
ces ;  Analysis  of  Ashes,  and  examination  of  Hair, 
Coins,  Arms  and  Stains,  as  applied  to  Chemical  Ju- 
risprudence, for  the  Use  of  Chemists,  Physicians, 
Lawyers,  Pharmacists  and  Experts  Translated  with 
additions,  including  a  list  of  books  and  Memoirs  on 
Texicology,  etc.  from  the  French  of  A.  Naquet  By 
J.  P.  Battershall,  Ph.  D.  with  a  preface  by  C.  F. 
Chandler,  Ph.  D.,  M.  D„  L.  L.  D.     i2mo,  cloth 

McCULLOCH.  Elementary  Treatise  on  the*  Mechan- 
ical Theory  of  Heat,  and  its  application  to  Air  and 
Steam  Engines.    By  R.  S.  McCulloch,  8vo,  cloth 

AXON.  The  Mechanics  Friend ;  a  Collection  of  Re- 
ceipts and  Practical  Suggestions  Relating  to  Aqua- 
ria— Bronzing — Cements — Drawing — Dyes —  Electri- 
city— Gilding — Glass  Working — Glues — Horology- 
Lacquers — Locomotives — Magnetism — Metal-Work- 
ing— Modelling— Photography— Pyrotechy — Railways 
— Solders— Steam  Engine— Telegraphy— Taxidermy 
—  Varnishes  —  Water-Proofing  and  Miscellaneous 
Tools, — Instruments,  Machines  and  Processes  con- 
nected with  the  Chemical  and  Mechanics  Arts ;  with 
numerous  diagrams  and  wood  cuts.  Edited  by  Wil- 
liam E.  A.  Axon.     Fancycloth 150 

17 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 

ERNST.  Manual  of  Practical  Military  Engineering,  Pre- 
pared for  the  use  of  the  Cadets  of  the  U .  S.  Military 
Academy,  and  for  Engineer  Troops.  By  Capt.  O.  H. 
Ernst,  Corps  of  Engineers,  Instructor  in  Practical 
Military  Engineering,  U.  S.  Military  Academy.  192 
wood  cuts  and  3  lithographed  plates.     i2mo,  cloth..     500 

BUTLER.  Projectiles  and  Rifled  Cannon.  A  Critical 
Discussion  of  the  Principal  Systems  of  Rifling  and 
Projectiles,  with  Practical  Suggestions  for  their  Im- 
provement, as  embraced  in  a  Report  to  the  Chief  of 
Ordnance,  U.  S.  A.  By  Capt.  John  S.  Butler,  Ord- 
nance Corps,  U.  S.  A.     36  plates,  4to,  cloth 7  50 

BLAKE.  Rejjort  upon  the  Precious  Metals :  Being  Sta- 
tistical Notices  of  the  principal  Gold  and  Silver  pro- 
ducing regions  of  the  World,  Represented  at  the 
Paris  Universal  Exposition.  By  William  P.  Blake, 
Commissionir  from  the  State  of  California.  8vo,  cloth    2  00 

TONER.  Dictionary  of  Elevations  and  Climatic  Regis- 
ter of  the  United  States.  Containing  in  addition  to 
Elevations,  the  Latitude,  Mean,  Annual  Temperature, 
and  the  total  Annual  iiain  fall  of  many  localities;  with 
a  brief  introduction  on  the  Orographic  and  Physical 
Peculiarities  of  North  America.  By  J.  M.  Toner, 
M.  D.     8vo,  cloth . .- 3  75 

MOWBRAY.  Tri-Nitro  Glycerine,  as  applied  in  the 
Hoosac  Tunnel,  and  to  Submarine  Blasting,  Torpe- 
does, Quarrying,  etc.  Being  the  result  of  six  year's 
observation  and  practice  during  the  manufacture  of 
five  hundred  thousand  pounds  of  this  explosive  Mica, 
Blasting  Powder,  Dynamites;  with  an  account  of  the 
various  Systems  of  Blasting  by  Electricity,  Priming 
Compounds,  Explosives,  etc.,  etc  By  George  M. 
Mowbray,  Operative  Chemist,  with  thirteen  illustra- 
tions, tables  and  appendix.  Third  Edition.  Re- 
written.   8vo- cloth 300 

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